Let $X$ be a Kahler variety with snc divisor $D$ such that $K_X+D$ is ample. then there is a Kahler metric $\omega_E$ such that $Ric(\omega_E)=-\omega_E$ on $E=X\setminus D$, then $h=\frac{1}{\omega_E^n}$, has not minimal singularity. But can we construct a singular hermitian metric so that has minimal singularities on pair $(X,D)$?

Note: We know from Demailly's theorem, every pseudoeffective line bundles on compact complex manifolds admit minimal singular metrics, but the Demailly's definition and Tsuji's definition are slightly different for minimal singularities. Also I am looking for singular hermitian metric which constructed from KE metric.

What about when $-(K_X+D)$ is ample? can we construct a singular hermitian metric so that is Analytical Zariski Decomposition with minimal singularities when we have log K-stability condition on pair $(X,D)$?